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Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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How interferometric techniques can be used to investigate the topology of polymers?

I need to study the topography of a specific polymer, knowing that my lab is an optics lab with many interferometric techniques, how can I use these techniques to investigate the topology?
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Simply Connected Spaces [on hold]

Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (...
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37 views

Topological shape of the equilibrium point

"All dynamical system possess topological shapes that characteristics it's equilibrium point"-so my question is what is the topological shape of the equilibrium point for a cart and Inverted pendulum ...
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84 views

What is the atomic limit?

I am attempting to grasp topological superconductivity for an assignment and in trying to understand what makes a quantum system topological have came across the following paragraph; "In the case ...
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1answer
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Chart(s) of space-time as a smooth manifold

So we all know that space-time in general relativity is modeled as a smooth (pseudoRiemannian) manifold. Each point (event) on space-time is labeled with a unique coordinate $(t,x,y,z)$ in a specific ...
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coulomb interaction on a ring

My qualitative understanding is that the mathematical form of the interaction between particles is constrained by their gauge symmetry, so that, for example, the U(1) gauge symmetry in QED gives rise ...
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Weyl spinor representations and the Lorentz group

I'm currently trying to read up on the Lorentz-group and it's representations. I've found a couple of posts here on stack-exchange that I find helpful and confusing at the same time, so I would be ...
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Metric of a cross-section in General Relativity [migrated]

Consider a finite closed region $V=(x,y,z)$ as a simply-connected subset of a 3-dimensional flat Euclidean space ${\Bbb R}^3$ with the metric $\text{d}s^2=\text{d}x^2+\text{d}y^2+\text{d}z^2$. A ...
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1answer
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Physical meaning of theorem

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
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Which mechanical system has a Moebius strip as its configuration space?

A harmonic oscillator has a line as its configuration space. A pendulum has a circle as its configuration space. Which system has a Moebius strip as its configuration space?
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The connection between symmetry and classifying spaces of a group

I recently read the following statement: "For any type of mathematical object, an object of that type with $G$ symmetry “is” a map from [its classifying space] $BG$ to the space of all objects ...
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1answer
93 views

Does the vierbein contain any extra information?

The vierbein from General relativity has $D(D+1)/2$ independent components when accounting for the $O(3,1)$ gauge symmetry. The metric has the same degrees of freedom. But does the vierbein contain ...
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Does a geodesic always extremize its path length? [duplicate]

I've learned that a geodesic maximizes its proper time in Minkowski spacetime. Is this still true in general curved spacetime? In other words, does the geodesic equation give the globally extremal ...
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What does it mean to define a spin-structure on a manifold? [closed]

I'm trying to think about what information I need to add to a manifold that it describes a spin structure? I know you can have spin-structure on a 2d plane, a 2-sphere. I also know you can define a ...
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The Simply Connected Manifold for $SU(3)$

$U(1)$ is the 1-sphere (S^1); $SU(2)$ is the 3-sphere (S^3); $SU(3)$ is _______________ (fill in the blank). What simply connected manifold is $SU(3)$ (isomorphic to)?
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Space/time movement in [closed]

Does time move in a seamless orientation, or can there be rips or tears in time that we do not know about? I'am asking out of curiosity only, not a study problem.
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The sphere $S^d$ is Euclidean space $E^d$ with infinity identified as a single point

I'm reading about anti de Sitter spacetime, and I found the following statement: $$ds^2 = \frac{1}{\cos^2 \psi} \big( -dt^2 + d\psi^2+ \sin^2 \psi d\Omega_{d-2}^2 \big).$$ Thus, the spatial ...
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Pure math courses for physicists: Topology [closed]

I'm in my bachelor in physics. In a couple of weeks I start my last year, and I'm interested in taking some pure math courses. As you see, I like the theoretical point of view, but I don't know if the ...
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A good instruction on Symmetry enriched Topological phases

I am looking for a good introduction to SETs, and topologically ordered phaeses it should be something describing first principles and gives a good explanation on the basics and the logic of this ...
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1answer
96 views

Reflection Vector (Ray Tracing)

Snell's law of refraction at the interface between 2 isotropic media is given by the equation: \begin{align} \tag{1} n_1 \,\text{sin} \,\theta_1 = n_2 \, \text{sin}\,\theta_2 \end{align} $\qquad$ ...
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2D BHZ tight binding model for Quantum spin Hall insulator

I am currently reading this article : https://arxiv.org/abs/cond-mat/0611341 and want to derive the k-space tight binding model of 2D BHZ. The tight binding model is written as \begin{equation} H = \...
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1answer
47 views

Is it possible understand Berry curvature as Gaussian curvature in some limit?

I would like to understand the Berry curvature and the Chern number from mathematical geometry-topology. I understand that in electronic QHE, there is a map from $k^2$ to a vector space where the ...
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1answer
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Spacetime has an infinite number of choices for differentiability. Coincidence?

Spacetime can be modelled using a four-dimensional topological manifold. Say we denote the manifold using $(M, \mathcal{O})$ where $\dim M =d$. The structure $(M,\mathcal{O})$ is not sufficient for ...
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1answer
101 views

How is Inflation able to create an infinite amount of energy?

Follow up to this question here: If the universe is flat, does that imply that the Big Bang produced an infinite amount of energy? As I understand Inflation theory, some time after the Big Bang, the ...
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2answers
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Clarification on statement in “Unitary Symmetry and Elementary Particles” by Lichtenberg

He says that: The set of values of the parameter or parameters which characterize a group element can be considered to be points in some kind of space. The number of parameters characterizes the ...
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About Witten's path integral formulation of Jones polynomial

In his landmark paper Quantum field theory and the Jones polynomial, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the ...
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1answer
51 views

Are fractional quantum hall effect system symetry enriched topological phases?

In the papers I review they first start to talk about topologically ordered phases of matter. Their standard example of it is FQHE. Than they give another set examples which are quantum spin liquids, ...
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2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where $$l_n = -z^{n+1} \partial_z$$ is an ...
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1answer
44 views

Is the quantum Hall state a topological insulating state?

I am confused about the quantum Hall state and topological insulating states. Following are the points (according to my naive understanding of this field) which confuse me: Topological insulator has ...
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1answer
34 views

Are interacting symmetry protected topological (SPT) phases and symmetry enriched topological (SET) phases must be gapped?

I wonder are interacting SPT and SET phases gaped? Can we have a SET or interacting SPT phase in a semi metal?
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1answer
51 views

Poincaré and Galilei group - notation

On this slide it just says that $\mathcal{P}$ and $\mathcal{G}$ are the Poincoré and Galilei groups, but I do not understand what they are made of. What does $\mathbb{R}^{1,3}$ mean? Why does $\...
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2answers
98 views

Spin statistics from the fundamental group of $SO(D)$

I read the answer to this question and am very intrigued by its simple and elegant explanation of the emergence of anyon, boson & fermion statistics. @Trimok basically says: In a space-time ...
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1answer
81 views

Rigorous definition of generalized coordinates

In Goldstein's classical mechanics and in many other books I haven't seen a rigorous definition of generalized coordinates. In a system of $N$ particles described by $\textbf{r}_1, \dots, \textbf{r}...
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1answer
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Configuration space of identical particles - fractional statistics

In Khare's book of fractional statistics and quantum theory, when discussing why we need fractional statistics he arrives at the configuration space for a system of two identical particles in $d$ ...
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1answer
87 views

Chern-Simons and framing dependence$.$

According to ref.1, the correlation functions of a Chern-Simons theory are topological invariants, up to the so-called framing, that is, the trivialisation of $TM\oplus TM$. The origin of this framing ...
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On the topology of the Einstein-Rosen bridge

I am currently working with Harvey Reall's notes on Black holes, and I have 2 questions concerning the interpretation of the Einstein-Rosen bridge. After some algebra, we are faced with the metric: $$...
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1answer
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How does extending a Chern-Simons theory to the bulk fix potential singularities?

According to ref.1 (§A.3), the naive definition of Chern-Simons $$ S[A]=k\int_M \mathrm{CS}[A]\tag{A.17} $$ is ill-defined, because $A$ may have "Dirac string singularities". The solution is to extend ...
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1answer
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Must the electromagnetic 2-form be harmonic in vacuum?

The Maxwell equations in vacuum are $dF=0$ and $d*F=0$. Is this not the same as saying $F$ is both closed and co-closed, and hence harmonic? But Hodge's theorem says the space of harmonic $p$-forms on ...
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1answer
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Recipe to determine symmetries of quadratic fermionic Hamiltonian in second quantisation

Consider an arbitrary 1D chain (of length $N$) of fermions with an arbitrary quadratic Hamiltonian of the form $$\mathcal{H}=\hat{\Psi}^\dagger H \hat{\Psi}$$ with $$\hat{\Psi}=\left(a_1, a_2, ...,...
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1answer
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Can solutions of GR have non-zero genus?

Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? If so, where can I read more about ...
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Nature of the elements of spacetime?

I am learning about relativity and am not quite sure how to think of spacetime. From a mathematical perspective, spacetime is a manifold i.e. a topological space for which about any point there exists ...
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1answer
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Why are two different gauge transformations of $A_\mu=0$ in $U(1)$ gauge thoery equivalent?

Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\...
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3answers
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Topological indices for systems that lack translational invariance

I have a 1D discrete, finite system that lacks translational invariance. It appears to have edge states, in much the same way as an SSH model has edge states. In the SSH model we can study the ...
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62 views

Can one find Dirac matrices for any spacetime metric?

For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{γ_μ,γ_ν\} = 2 g_{μν} I?$$ For a diagonal metric I was able to work out that $$\bar{γ}_μ=\sqrt{n_{μμ}*...
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2answers
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Is the classification of (Symetry Protected) Topological Order for 3 band models different than for two band models?

I was reading this article: https://arxiv.org/abs/1512.08882 on the 10 fold way which gives a nice explanation of the possible topological phases for each of the symmetry classes. The example ...
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1answer
71 views

What is the topology of a phase diagram?

Looking at various two-variable phase diagrams I was struck by that on every one I have seen so far all the phases formed simple connected regions; see, for example the phase diagrams of $H_2O$ or of $...
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1answer
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What is direct interaction, if exist, between gluons and pions?

Gluons mediate the strong force between quarks. Pions mediate the nuclear force or nucleon-nucleon interaction or residual strong force. I had thought of some scalar bosons for my idea because if I'm ...
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78 views

If the universe is a $3$-torus, must it be longer in some direction?

So in a finite universe, I've read one possible topology for a flat universe is a 3 torus. On a 2 torus, it's obviously longer in one direction than the other. Would the same hold true for the ...
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Conformal compactification of spacetimes

I’m looking answers and/or references concerning the following questions about conformal compactification. Given a $d$-dimensional spacetime $(M,g)$ (or simply just for $d=4$): Does the conformal ...